SISSA Geometry Seminars

[Ugo Bruzzo]

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SISSA GEOMETRY SEMINARS
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Ada Boralevi (SISSA)

Title: Spaces of matrices of constant rank and instantons

Wednesday, January 9th, 4:30 pm in Room 136

Abstract: Given a complex vector space V of dimension n, one can look
at d-dimensional linear subspaces A in \Wedge^2(V), whose elements
have constant rank r. The natural interpretation of A as a vector
bundle map yields some restrictions on the values that r,n and d can
attain. After a brief overview of the subject and of the main
techniques used, I will concentrate on the case r=n-2 and d=4. I will
introduce what used to be the only known example, by Westwick, and
give an explanation of this example in terms of instanton bundles and
the derived category of P^3. I will then present a new method that
allows to prove the existence of new examples of such spaces, and show
how this method applies to instanton bundles of charge 2 and 4. These
results are in collaboration with D.Faenzi and E.Mezzetti.

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Giovanni Marelli

Universidad de Antioquia, Medellin, Colombia

Title: Gradient-like vector fields on a complex analytic variety

Thursday, January 10th, 4:30 pm in Room 136

Abstract: Given a complex analytic function $f$ on a Whitney stratified complex analytic variety of complex dimension $n$, whose real part $Re(f)$ is Morse, we prove the existence of a stratified gradient-like vector field for $Re(f)$ such that the unstable set of a critical point $p$ on a stratum $S$ of complex dimension $s$ has real dimension $m(p)+n-s$, where $m(p)$ is the Morse index of the restriction of $f$ to $S$, as was conjectured by Goresky and MacPherson. We expect as application the construction of the Morse-Witten complex for intersection homology.

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U. Bruzzo